# 5 5 duoprism

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In geometry of 4 dimensions, a 5-5 duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons.

## Contents

It has 25 vertices, 50 edges, 35 faces (25 squares, and 10 pentagons), in 10 pentagonal prism cells. It has Coxeter diagram , and symmetry [[5,2,5]], order 200.

## Images

Seen in a skew 2D orthogonal projection, 20 of the vertices are in two decagonal rings, while 5 project into the center. The 5-5 duoprism here has an identical 2D projective appearance to the 3D rhombic triacontahedron. In this projection, the square faces project into wide and narrow rhombi seen in penrose tiling.

## Related complex polygons

The regular complex polytope 5{4}2, , in C 2 has a real representation as a 5-5 duoprism in 4-dimensional space. 5{4}2 has 25 vertices, and 10 5-edges. Its symmetry is 5[4]2, order 50. It also has a lower symmetry construction, , or 5{}×5{}, with symmetry 5[2]5, order 25. This is the symmetry if the red and blue 5-edges are considered distinct.

## Related honeycombs and polytopes

The birectified order-5 120-cell, , constructed by all rectified 600-cells, a 5-5 duoprism vertex figure.

## 5-5 duopyramid

The dual of a 5-5 duoprism is called a 5-5 duopyramid. It has 25 tetragonal disphenoid cells, 50 triangular faces, 35 edges, and 10 vertices.

It can be seen in orthogonal projection as a regular 10-gon circle of vertices, divided into two pentagons, seen with colored vertices and edges:

## Related complex polygon

The regular complex polygon 2{4}5 has 10 vertices in C 2 with a real represention in R 4 matching the same vertex arrangement of the 5-5 duopyramid. It has 25 2-edges corresponding to the connecting edges of the 5-5 duopyramid, while the 10 edges connecting the two pentagons are not included. The vertices and edges makes a complete bipartite graph with each vertex from one pentagon is connected to every vertex on the other.

## References

5-5 duoprism Wikipedia

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